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17   APPLICATIONS: II DUAL OF L1         201

17. APPLICATIONS: II. DUAL OF Ll

(13.17.1 ) Let y. be a positive measure on X. For each function g e ^f^(X, u)
(resp. g e ^^(X, /*)), the mappingdg:f\-* \fg dp, is a continuous linear form

on the space JSf i(X, u) (resp. JS?c(X> u)) with respect to the topology defined by
the norm Nx. The corresponding continuous linear form 9g: f\ > ti(fg) on the
Banach space L^(X, u) (resp. L(X, a)) has norm (5.7.1) equal to N00(^).
Conversely, every continuous linear form on J2?i(X, u) (resp. cSfc(X, u)) is
of the form f ^r^i fg du, where g is a function belonging to JzfP(X, u) (resp.
^^(X, //)), and the equivalence class of g is uniquely determined.

We shall give the proof for JSf^  The fact that 6g is a continuous linear form
on J*?c and the inequality \\Sg\\ ^ N^^), are consequences of (13.12.5). To
prove that \\8g\\ = N ,(#), we may assume that g is not /^-negligible (in other
words, that N00(^) > 0), since the assertion is trivial otherwise.

Let a be any real number such that 0 < a < N00(^). Then there exists a
compact subset K, of X, with measure /^(K) > 0, such that g \ K is con-
tinuous and \g(x)\ > a for all x e K ((13.9.9) and (13.9.4)). For each a > 0
there exists a finite covering (U) of K by open sets in K, such that the oscilla-
tion of g in each Uf is ^8 (3.16.5). Hence there exists a partition of K into a
finite number of integrable sets Ak such that the oscillation of g in each Afc
is ^e (13.9.12.1). At least one of these sets, which we shall denote by A, has
measure u(A) > 0. Let b be one of the values taken by g in A. We have
\b\ > a and \g(x)  b\ :g e for all x e A. Now put

clearly fe ?lc and Nx(/) = 1. Also we have

f 5                                  f 5

 gcpA dp.  =   \b\u(A) +     (g - b)(pA dp.
J PI                                  J \b\

so  that    |/^rfjU ^.a  e. This proves  our assertion, because a>0 and

a < NOO(#) were arbitrary.

Conversely, let u be a continuous linear form on 3?^. Then there exists a
real number c> 0 such that \u(f)\ g c - N^/) for all/6 &lc (12.14.1). If K is
any compact subset of X, it is known (13.11.6) that the topology induced on
Jf C(X; K) by that of J$?c(x> A*) is coarser than the topology defined by the
norm ||/||. Hence the linear form u is continuous on each of the Banachis another positive measure on X